$\frac{1}{1-x} = \sum_{i \geq 0} x^i$, when $|x| < 1$.
Multiplying both sides by $x$, we get -
$\frac{x}{1-x} = \sum_{i \geq 1} x^i$, when $|x| < 1$.
Now,
$\frac{3}{1+5x} = 3 \times \sum_{i \geq 0} (-5x)^i \implies$ Coefficient of $x^n = 3 \times (-5)^n$.
$\frac{-2}{7-2x} = \frac{-2}{7} \times \sum_{i \geq 0}(\frac{2x}{7})^i \implies$ Coefficient of $x^n = \frac{-2}{7} \times (\frac{2}{7})^n$.
$\frac{5x}{3+2x} = \frac{5}{3} \times \sum_{i \geq 1}(\frac{-2x}{3})^i \implies$ Coefficient of $x^n = \frac{5}{3} \times (\frac{-2}{3})^n$.
$\frac{7x}{5-2x} = \frac{7}{5} \times \sum_{i \geq 1}(\frac{2x}{5})^i \implies$ Coefficient of $x^n = \frac{7}{5} \times (\frac{2}{5})^n$.
Final answer,
Coefficient of $x^n = 3 \times (-5)^n - \frac{2}{7} \times (\frac{2}{7})^n + \frac{5}{3} \times (\frac{-2}{3})^n + \frac{7}{5} \times (\frac{2}{5})^n$
